The analysis of smooth games has attracted attention, motivated by the success of adversarial formulations. The Hamiltonian method is a lightweight second-order approach that recasts the problem in terms of a minimization objective. Consensus optimization can be seen as a generalization: it mixes a Hamiltonian term with the original game dynamics. This family of Hamiltonian methods has shown promise in literature. However, they come with no guarantees for stochastic games. Classic stochastic extragradient and mirror-prox methods require averaging over a compact domain to achieve convergence. Recent variance-reduced first-order schemes focus on unbounded domains, but stop short of proving last-iterate convergence for bilinear matrix games. We analyze the stochastic Hamiltonian method and a novel variance-reduced variant of it and provide the first set of last-iterate convergence guarantees for stochastic unbounded bilinear games. More generally, we provide convergence guarantees for a family of stochastic games, notably including some non-convex ones. We supplement our analysis with experiments on a stochastic bilinear game, where our theory is shown to be tight, and simple adversarial machine learning formulations.